These slides are intended to get you thinking about the data you are using, what it tells us, what it doesn’t etc., and to think carefully about what we’re trying to measure, and what we’re actually measuring.
# anscombes quartet
longanscombe <- anscombe %>%
pivot_longer(cols = everything(), #pivot all the columns
cols_vary = "slowest", #keep the datasets together
names_to = c(".value", "set"), #new var names; .value=stem of vars
names_pattern = "(.)(.)") #to extract var names
kable(
list(anscombe),
caption="Anscombe's Quartet",
booktabs = TRUE,
valign = 't',
row.names = FALSE,
)
|
B <- data.frame(set=numeric(0), b0=numeric(0), b1=numeric(0))
for (i in longanscombe$set) {
m <- lm(y ~ x, data=longanscombe %>% filter(set==i))
B[i:i,] <- data.frame(i, coef(m)[1], coef(m)[2])
}
kable(
list(B),
caption="Anscombe's Quartet",
booktabs = TRUE,
valign = 't',
row.names = FALSE,
)
|
ggplot(longanscombe, aes(x = x, y = y)) +
geom_point() +
facet_wrap(~set) +
geom_smooth(method = "lm", se = FALSE, color="red")
The data generating process is the complete description of how the observed data arose and how other such data would arise. It includes variables, conditionals, functional forms, mappings from one unit to another, etc.
What are the units of observation? Who is taking action or having action done to them? Are the units heterogeneous, and if so, how?
What circumstances are the units in?
What are the units’ choice sets?
What relates the units circumstances to the outcomes?
How are the units related to each other?
What don’t we observe?
| Outcome | Autocrat | Democrat | Total |
|---|---|---|---|
| Loser | 42 | 9 | 51 |
| Winner | 32 | 38 | 70 |
| Total | 74 47 | 121 |
From Lake (1992), p. 31
A type III error, introduced in Kimball (1957), occurs when a researcher produces the right answer to the wrong question. A corollary of this rule, as noted by Chatfield (1995, p. 9), is that an approximate answer to the right question is worth a great deal more than a precise answer to the wrong question.
Collections of alike units, their characteristics, features, choice sets, behaviors, etc.
Variables are either
discrete - observations match to integers; all possible values are clearly distinguishable; not divisible. E.g., number of protests in DC this year; an individual’s sex; Polity score.
continuous - observations can take on any real value between boundaries (sometimes $-, +); infinitely divisible. E.g., household income, GDP per capita.
May be of two types or levels of measurement:
nominal - categories are distinct, but lack order. E.g., religion = Hindu, Muslim, Catholic, Protestent, Jewish. Binary variables are nominal, e.g., Sex = male (0), female (1); do you have blue eyes? yes (0), no (1).
ordinal - take on countable values, increasing/decreasing in some dimension. E.g., Polity -10, -9, \(\ldots\) 0, 1, \(\ldots\) 9, 10 increasing in democracy; survey responses “Do you feel safe traveling abroad?” Not at all; sometimes; yes, completely.
Can be of two types (levels of measurement):
interval - 1 unit increase has same meaning across the scale (i.e., the intervals are the same); e.g., degrees Celsius or Fahrenheit.
ratio - intervals but also has a meaningful absolute zero; e.g., weight in pounds; zero lbs indicates the absence of weight; Venmo balance = zero, means actually no money; degrees Kelvin. Duration of a war in days - zero days means there’s no war.
These four levels or measurement can be ordered by the amount of information a variable contains:
nominal
ordinal
interval
ratio
We can turn higher levels to lower levels, but not the opposite - doing so sacrifices information.
In general, the level of measurement of \(y\) (so the type and amount of information in a variable) shapes what type of model is appropriate.
discrete variables usually require statistics/models in the Binomial family (for our purposes, mostly MLE models like the Logit.)
continuous variables usually require statistics/models in the Normal/Gaussian family (for our purposes, mostly OLS models like the linear regression.)
Continuous or discrete; what can you say about these data from their observed distribution?
gap <- read.csv("/Users/dave/documents/teaching/501/2024/slides/L1-data/data/gapminder.csv")
ggplot(gap, aes(x=alcohol_consumption_per_adult_15plus_litres)) +
geom_histogram(aes(y=..density..), colour="black", fill="white")+
geom_density(alpha=.2, fill="#FF6666") +
labs(x = "Liters of Booze", y= "Density", caption="Alcohol Consumption, Gapminder")+
ggtitle("Density - Alcohol Consumption per Adult (Liters)") 
gap$nalc <- dnorm(gap$alcohol_consumption_per_adult_15plus_litres, mean=mean(gap$alcohol_consumption_per_adult_15plus_litres, na.rm = TRUE), sd=sd(gap$alcohol_consumption_per_adult_15plus_litres, na.rm = TRUE))
ggplot() +
geom_histogram(data=gap, aes(x=alcohol_consumption_per_adult_15plus_litres, y=..density..), colour="black", fill="white") +
geom_density(data=gap, aes(x=alcohol_consumption_per_adult_15plus_litres), alpha=.2, fill="#FF6666") +
geom_line(data=gap, aes(x=alcohol_consumption_per_adult_15plus_litres, y=nalc), linetype="longdash", size=1)+
labs(x = "Liters of Booze", y= "Density", caption="Alcohol Consumption, Gapminder")+
ggtitle("Alcohol Consumption per Adult (Liters), Normal PDF")
polity <- polity %>% mutate(era = ifelse(year==1980, 1980, ifelse(year==2018, 2018, 0)))
p <- ggplot(data=polity %>% filter(era!=0), aes(y=polity2), colour="black", fill="white")+
geom_bar()+
geom_text(aes(label = ..count..),stat="count", hjust = -.2, colour = "black", size=2.5,
position = position_dodge(0.9))
p + facet_wrap(era ~ .) +
labs(y = "Polity score", x= "Countries", caption="Polity project")+
ggtitle("Polity Scores, 1980 and 2018") 
Two variables - Polity, and the number of efforts (successful, unsuccessful) by state leaders to overstay their terms in office.
#polity <- read_dta("/Users/dave/Documents/2023/PITF/slides/polity5.dta")
overpolity <- left_join(tl, polity, by=c("ccode"="ccode", "year"="year"))
success <- overpolity %>%
group_by(polity2) %>%
summarise_at(c("tl_success"), sum) %>%
filter(!is.na(polity2)) %>%
mutate(outcome = "succeeded") %>%
mutate(events=tl_success)%>%
subset(select = -c(tl_success))
fail <- overpolity %>%
group_by(polity2) %>%
summarise_at(c("tl_failed"), sum) %>%
filter(!is.na(polity2)) %>%
mutate(outcome = "failed") %>%
mutate(events=tl_failed) %>%
subset(select = -c(tl_failed))
tlpolity <- rbind(success, fail)
ggplot(tlpolity, aes(fill=outcome, y=events, x=polity2)) +
geom_bar(position="stack", stat="identity",)+
scale_fill_manual(values=c("dark green", "light green")) +
labs(x = "Polity", y= "Frequency", caption="Overstaying data (Versteeg et al. 2020)") +
ggtitle("Overstay Attempts over Regime") +
scale_x_continuous(breaks=seq(-10, 10, 1))